Centers for the Kukles homogeneous systems with even degree

Giné, Jaume; Llibre, Jaume; Valls, Claudia

doi:https://doi.org/10.11948/2017093

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2017

Author

Giné, Jaume

Llibre, Jaume

Valls, Claudia

Suggested citation

Giné, Jaume; Llibre, Jaume; Valls, Claudia; . (2017) . Centers for the Kukles homogeneous systems with even degree. Journal Of Applied Analysis And Computation, 2017, vol. 7, núm. 4, p. 1534-1548. https://doi.org/10.11948/2017093.

Abstract

For the polynomial differential system x˙=−y, y˙=x+Qn(x,y), where Qn(x,y) is a homogeneous polynomial of degree n there are the following two conjectures done in 1999. (1) Is it true that the previous system for n≥2 has a center at the origin if and only if its vector field is symmetric about one of

the coordinate axes? (2) Is it true that the origin is an isochronous center of the previous system with the exception of the linear center only if the system has even degree? We give a step forward in the direction of proving both conjectures for all n even. More precisely, we prove both conjectures in the case n=4 and for n≥6 even under the assumption that if the system has a center or an isochronous center at the origin, then it is symmetric with respect to one of the coordinate axes, or it has a local analytic first integral which is continuous in the parameters of the system in a neighborhood of zero in the parameters space. The case of n odd was studied in [8].

URI

http://hdl.handle.net/10459.1/60424

DOI

https://doi.org/10.11948/2017093

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Journal Of Applied Analysis And Computation, 2017, vol. 7, núm. 4, p. 1534-1548

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Except where otherwise noted, this item's license is described as cc-by (c) Giné et al., 2017